Abstract. The Apollonian metric is a generalization of the
hyperbolic metric, defined in a much larger class of open sets.
However, since it was introduced by Beardon
in 1998, it has remained an open question what its isometries are.
Beardon first raised this question and asked if the Apollonian
isometries were just Möbius mappings.
In this paper we show that this is the case in open sets with regular,
for instance C1, boundary.